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Fuzzy Sets and Many-Valued Logics

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Quantum Physics, Fuzzy Sets and Logic

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Abstract

Classical, two-valued logic is a basis of traditional mathematics and, in particular, of the traditional set theory. Although well-elaborated systems of axioms for the classical set theory do exist, for all practical purposes it is enough to distinguish a set that we are interested in by a predicate which, according to two-valued logic, enable all the objects under consideration to be unambiguously divided into two disjoint classes: objects that belong to a set and objects that do not belong to a set and form its complement.

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Notes

  1. 1.

    It seems that Chang [2] and Klaua [3] elaborated similar ideas independently of [1] and published them even slightly before [1]. It is for historians of science to explain why their papers, contrary to [1] are almost neglected.

  2. 2.

    This observation gave rise to the notion of probabilistic fuzzy sets introduced by Hirota [4], whose membership functions are themselves “fuzzy”. Of course this procedure can be continued, but objects obtained in this way are less and less convenient to deal with.

  3. 3.

    There are also operations which can be defined on fuzzy sets that have no counterparts in traditional set theory, e.g. the operation of “sharpening” a set which makes it “less fuzzy”.

  4. 4.

    Except almost neglected papers [2, 3] mentioned in footnote 1 of this chapter.

  5. 5.

    It is clear that Giles [5] was not aware of Zawirski’s 1934 paper [6] where these operations appeared for the first time (in the domain of a many-valued logic), and he was also, most probably, not aware of Frink’s 1938 paper [7]. It also seems that these operations were rediscovered many times by various authors which explains the multiplicity of their names. Although these operations did not appear explicitly in any Łukasiewicz paper, the name Łukasiewicz operations seems to be the most popular nowadays and will be used throughout this paper.

  6. 6.

    There are \(2^{(2^2)}=16\) conceivable two-argument connectives in 2-valued logic, \(3^{(3^2)}=19.683\) two-argument connectives in 3-valued logic, \( n^{(n^2)}\) two-argument connectives in n-valued logic and obviously infinity of two-argument connectives in infinite-valued logic. Of course, not all of these could, in a reasonable way, be interpreted as a disjunction or a conjunction. Some of them could be interpreted as an implication or equivalence, but the overwhelming majority surely has no two-valued counterparts.

  7. 7.

    Even the whole families of such operations, parametrized by real numbers, have already been studied (see, e.g. [8, 9]).

References

  1. Zadeh, L. A. “Fuzzy sets”, Information and Control, 8 (1965) 338–353.

    Google Scholar 

  2. Chang, C. C. “Infinite valued logics as a basis of set theory”, Proceedings of the 1964 International Congress in Logic, Methodology, and Philosophy of Science (North-Holland, Amsterdam, 1965) pp. 93–100.

    Google Scholar 

  3. Klaua, D. “Über einen Einsatz zur mehrwertigen Mengenlehre”, Monatsb. Deutsch. Acad. Wiss. Berlin, 7 (1965) 859–867.

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  4. Hirota, M. “Concepts of probabilistic sets”, Proceedings of IEEE Conference on Decision and Control (1977) 1361–1366; Fuzzy Sets and Systems, 5 (1981) 31–46.

    Google Scholar 

  5. Giles, R. “Łukasiewicz logic and fuzzy set theory”, International Journal of Man-Machine Studies, 67 (1976) 313–327.

    Google Scholar 

  6. Zawirski, Z. “Jan Łukasiewicz 3-valued logic. On the logic of L. E. J. Brouwer. Attempts at applications of many-valued logic to contemporary natural science”, Sprawozdania Poznańskiego Towarzystwa Przyjaciół Nauk, 2–4 (1931) 1–8 (in Polish).

    Google Scholar 

  7. Frink, O. “New algebras of logic”, American Mathematics Monthly, 45 (1938) 210–219.

    Google Scholar 

  8. Frank, M. J. “On the simultaneous associativity of \(F(x,y)\) and \(x+y-F(x,y)\)”, Aequationes Mathematicae, 19 (1979) 194–226.

    Google Scholar 

  9. Yager, R. R. “On general class of fuzzy connectives”, Fuzzy Sets and Systems, 4 (1986) 235–242.

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, University of Gdańsk, Gdańsk, Poland

    Jarosław Pykacz

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  1. Jarosław Pykacz
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Pykacz, J. (2015). Fuzzy Sets and Many-Valued Logics. In: Quantum Physics, Fuzzy Sets and Logic. SpringerBriefs in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-19384-7_4

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